Linux Mint, currently, is the most downloaded Linux distro, at least according to Distrowatch.com. Ubuntu used to be the most downloaded Linux until they came out with that Unity interface. My brother currently uses Linux Mint (an old version, 10) and he hasn't had many problems with it. The only issues, minor ones, is that Firefox sometimes freezes on loading pictures and he can't play Minecraft.
Wine is pretty easy to use. Once you have it installed, all you have to do is download the Windows program you want to use, rightclick and check the option to run it as an executable and save it. Then right click the same file again and go to Open With, and select the Wine Program Loader. This should install it as if it's on Windows.
There's also PlayOnLinux, a companion of Wine, which you can also use to install Windows stuff, though you can only pick from a list. It has a ton of games with it.